Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n, tc... Th dpndnc on t R will b omittd in th notation. 9. Wak asymptotics Writ π n.n = n j= (z η j,n), so that η j,n ar th zros of π n,n. W will show that for t 2 ths zros accumulat on a singl intrval [ a, a] and hav dnsity dµ V (x) = x2 (c + π 2 ) a 2 x 2, on [ a, a], whr a = c = t+ t 2 /4+3 2t+2 t 2 +2 3 3 On can show that with ths constant, th total mass of th masur is, i.. a a dµ V (x) =.
2 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I For intrprting th following thory w not that π n,n(z) = n π n,n n n j= = z η j,n z x dµ n(x). whr µ = n n j= δ η j,n is th normalizd sum of dirac masss at th zros of π n,n. Thorm 9... Lt t 2. Thn π n,n(z) lim = n n π n,n a a z x dµ V (x), uniformly for z in compact substs of C \ [ a, a]. W will prov this Thorm, among strongr rsults, by using Rimann- Hilbrt tchniqus. In this lctur w will always assum Th cas t = 2 will b don latr. 9.2 Th g-function t > 2. Dfinition 9.2.. W dfin g : C \ (, a] as g(z) = a a log(z x)dµ V (x), whr w choos th branch of th logarithm such that g is analytic and ral for z > a. Lmma 9.2.2. g (z) = V (z) 2 (c + z 2 /2)(z 2 a 2 ) /2, whr tak th z (z 2 a 2 ) /2 such that it is analytic in C \ [ a, a] and positiv on (a, ). Proof. Dfin th function h = Cauchy transform of dµ V (x), i.. h(z) = 2i 2i g. a a Thn h is, is up to constant, th x z dµ V (x).
9.2. THE G-FUNCTION 3 But that mans that h is th uniqu solution to th following RHP (xrcis!) h is analytic in C \ [ a, a] h + (x) h (x) = (c + x2 2 ) a 2 x 2, x ( a, a), h(z) = O(/z), z h is boundd nar ± a. W claim that 2i (V (z)/2 (c + z 2 /2)(z 2 a 2 ) /2) solvs th sam RHP problm and hnc,by uniqunss of th solution, w provd th statmnt. To s that this indd solv th RHP w nd to chck th jump condition, which follows by th choic of th squar root (xrcis), and th asymptotic bhavior at infinity. Aftr a computation on chcks that V (z) cancls th polynomials part of th scond trm and hnc th claim follows. Th following proprtis will b crucial in th upcoming analysis. Lmma 9.2.3. Th function g satisfis th following proprtis g + (x) g (x) = 2πi for x < a g + (x) g (x) = 2πi a x for x [ a, a]. g + (x) + g (x) V = l for x [ a, a] g + (x) + g (x) V < l for x R \ [ a, a]. for som constant l R. Proof.. Not that For z, x R w hav log ± (z x) = log z x + i arg ± (z x) arg + (z x) arg (z x) = Sinc dµ V =, w find th statmnt. 2. Follows by th sam argumnts as. { 2πi, z < x 0, z x.
4 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I 3. This follows form th prvious lmma. By th dfinition of th squar root w hav (z 2 a 2 ) /2 ± = ±i a 2 z 2 for z ( a, a) and hnc g +(x) + g (x) V (x) = 0 for x [ a, a]. 4. This follows form th sam argumnts as bfor but now w hav g +(x) + g (x) V (x) < 0 on (a, ) on (, a). g +(x) + g (x) V (x) > 0 9.3 Ovrviw of th Dift/Zhou stpst dscnt analysis W start with th RHP for th orthogonal polynomials as discussd int h prvious lctur (with w(x) = nv (x) ). RH problm 9.3.. W sk for a function Y : C \ R C 2 2 such that Y is analytic in C \ R. ( ) nv (x) Y + (x) = Y (x), for x R. 0 ( ) z n 0 Y (z) = (I + o()) 0 z n as z. Th asymptotic analytis consists of a squnc of transformations Y T S R. Each transformation is simpl and invrtibl. Aftr ach transformation w obtain a nw RiHP. In th nd, th goal is to nd up with a RHP of th form { R + = R J r R(z) = I + O(/z), z, with J R as n. W thn find R I and morovr, an asymptotic xpansion for R. By tracing th transformations back, w obtain an asymptotic xpansion for Y.
9.4. FIRST TRANSFORMATION : NORMALIZING THE RHP NEAR INFINITY5 9.4 First transformation : normalizing th RHP nar infinity In th first transformation ( ) ( ) nl/2 0 n(g l/2) 0 T (z) = 0 nl/2 Y (z) 0 n(g l/2). Thn T satisfis RHP that w will now pos. First of all, w not that sinc µ V has total mass w hav g(z) = log z + O(/z), z. That mans that T (z) = I + O(/z), which mans that w hav normalizd th RHP at infinity. Of cours, th pric w pay is in trms of a mor complicatd jump structur which w can comput from ( ) ( ) J T = T T n(g l/2) 0 + = 0 n(g Y l/2) Y n(g l/2) 0 + 0 n(g l/2). A straightforward computation thn givs that T solvs th following RHP. RH problm 9.4.. W sk for a function T : C \ R C 2 2 such that T is analytic in C \ R. ( n(g + (x) g (x)) T + (x) = T (x) n(g ) +(x)+g (x) V (x) l) 0 n(g +(x) g (x)), for x R. T (z) = I + O(/z) as z. Thr is a simplification for th jump matrix for T by th proprtis of th g-function in Lmma 2.2.3. Thn ( n(g + (x) g (x)) n(g ) +(x)+g (x) V (x) l) 0 n(g +(x) g (x)) ( ) n(g +(x) g (x)) 0 n(g, x [ a, a] +(x) g (x)) = ( ) n(g +(x)+g (x) V (x) l), x R \ [ a, a]. 0 (9.4.)
6 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I ( ) nϕ a ( nϕ + ) ( ) nϕ 0 0 nϕ a 0 Figur 9.: Jumps for T W will introduc som notation for prsntation purposs. W dfin, for z C \ (, a] ϕ(z) = 2 z a (c + y2 2 )(y2 a 2 ) /2 dy, whr th path of intgration is C \ (, a]. Thn ϕ is analytic. Morovr, ϕ + ϕ = 0 mod 2πi on (, a] and thrfor w hav that nϕ is analytic in C \ [ a, a]. Morovr, by Lmmas 2.2.2 and 2.2.3 w s that w can rwrit th jump in trms of ϕ and obtain ( ) nϕ + T + = T 0 nϕ on [ a, a] and nϕ ( T + = T 0 ) on R \ [ a, a]. S also Figur??. Sinc R ϕ > 0 on R \ [ a, a] w s that th jumps on R \ [ a, a] ar xponntially small in n. Howvr, th jumps on [ a, a] ar not xponntially small. In fact, ϕ ± (x) = ±2πi a x dµ V (x) and hnc th jumps on [ a, a] ar highly oscillating! 9.5 Scond Transformation: opning of th lnss To liminat th highly oscillating jumps w prform a transformation that is known as th opning of th lns. It is basd on th following factorization ( ) ( ) ( ) ( ) nϕ + 0 0 0 0 nϕ = nϕ 0 nϕ +. Now w do th following. W opn up a lns around th intrval [ a, a]. That is, w tak two simpl contours conncting a to +a, on in th uppr
9.5. SECOND TRANSFORMATION: OPENING OF THE LENSES 7 half plan and on in th lowr half plan. W thn dfin ( ) 0 T nϕ in uppr part ( ) S = 0. T nϕ in lowr part T lswhr. S also Figur 2.2. Now what did w gain from this transformation. Proposition 9.5.. For t > 2 w can choos th lips of th lns such that R ϕ < 0 on th lipss. Proof. Not that ϕ is analytic in C \ (, a] and, for x [ a, a], ϕ + (x) = 2πi a x dµ V (x). Bcaus of th Cuachy Rimann quations (not that ϕ can analytically continud to [ a, a]) w hav with z = x + iy d R ϕ dy (x) = d Im ϕ dx (x) = dµ V dx (x) > 0 (sinc t > 2). But that mans that w can indd dform th intrval [ a, a] to a path from a to a in th uppr half plan such that R ϕ < 0 on that path. Th argumnt for th lowr lips is analogous. In Figur 2.4 w show th part of th complx plan whr R ϕ < 0 for t =. Clarly, th lns can b chosd such that it falls compltly in th rgion. Conclusion: all jumps for S othr thn th jump on [ a, a] ar xponntially small in n (pointwis). Th jump on [ a, a] dos not dpnd on n. Hnc w hav mad big progrss in th undrstanding of th asymptotic bhavior of Y.
8 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I S = T S = T S = T ( ) 0 nϕ ( ) 0 nϕ Figur 9.2: Opning of th lns ( ) 0 nϕ ( ) nϕ 0 ( 0 ) 0 ( ) nϕ 0 ( ) 0 nϕ Figur 9.3: Jumps for S
9.5. SECOND TRANSFORMATION: OPENING OF THE LENSES 9 3 2 0 2 3 3 2 0 2 3 Figur 9.4: Th whit rgion in which is th rgion whr R ϕ < 0 for t =. Th shadd rgion is th rgion R ϕ > 0.
0 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I
Bibliography [] M.J. Ablowitz and A.S. Fokas, Complx variabls: introduction and applications. Scond dition. Cambridg Txts in Applid Mathmatics. Cambridg Univrsity Prss, Cambridg, 2003. [2] K. F. Clancy and I. Gohbrg, Factorization of matrix functions and singular intgral oprators. Oprator Thory: Advancs and Applications, 3. Birkhusr Vrlag, Basl-Boston, Mass., 98 [3] P.A. Dift,Orthogonal polynomials and random matrics: a Rimann- Hilbrt approach. Courant Lctur Nots in Mathmatics, 3. Nw York Univrsity, Courant Institut of Mathmatical Scincs, Nw York; Amrican Mathmatical Socity, Providnc, RI, 999. viii+273 pp. [4] A.S. Fokas, A. Its, A.A. Kapav and V.Y. Novokshnov, Painlv Transcndnts. Th Rimann-Hilbrt approach. Mathmatical Survys and Monographs, 28. Amrican Mathmatical Socity, Providnc, RI, 2006. xii+553 pp. [5] I. Gohbrg and N. Krupnik, On-dimnsional linar singular intgral quations. I. Introduction. Oprator Thory: Advancs and Applications, 53. Birkhusr Vrlag, Basl, 992.